\(\int x^2 (\pi +c^2 \pi x^2)^{5/2} (a+b \text {arcsinh}(c x)) \, dx\) [82]

Optimal result

Integrand size = 26, antiderivative size = 212 \[ \int x^2 \left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=-\frac {5 b \pi ^{5/2} x^2}{256 c}-\frac {59}{768} b c \pi ^{5/2} x^4-\frac {17}{288} b c^3 \pi ^{5/2} x^6-\frac {1}{64} b c^5 \pi ^{5/2} x^8+\frac {5 \pi ^2 x \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{128 c^2}+\frac {5}{64} \pi ^2 x^3 \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))+\frac {5}{48} \pi x^3 \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {1}{8} x^3 \left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))-\frac {5 \pi ^{5/2} (a+b \text {arcsinh}(c x))^2}{256 b c^3} \] Output:

-5/256*b*Pi^(5/2)*x^2/c-59/768*b*c*Pi^(5/2)*x^4-17/288*b*c^3*Pi^(5/2)*x^6- 
1/64*b*c^5*Pi^(5/2)*x^8+5/128*Pi^2*x*(Pi*c^2*x^2+Pi)^(1/2)*(a+b*arcsinh(c* 
x))/c^2+5/64*Pi^2*x^3*(Pi*c^2*x^2+Pi)^(1/2)*(a+b*arcsinh(c*x))+5/48*Pi*x^3 
*(Pi*c^2*x^2+Pi)^(3/2)*(a+b*arcsinh(c*x))+1/8*x^3*(Pi*c^2*x^2+Pi)^(5/2)*(a 
+b*arcsinh(c*x))-5/256*Pi^(5/2)*(a+b*arcsinh(c*x))^2/b/c^3
 

Mathematica [A] (verified)

Time = 0.61 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.92 \[ \int x^2 \left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {\pi ^{5/2} \left (2880 a c x \sqrt {1+c^2 x^2}+22656 a c^3 x^3 \sqrt {1+c^2 x^2}+26112 a c^5 x^5 \sqrt {1+c^2 x^2}+9216 a c^7 x^7 \sqrt {1+c^2 x^2}-1440 b \text {arcsinh}(c x)^2+576 b \cosh (2 \text {arcsinh}(c x))-144 b \cosh (4 \text {arcsinh}(c x))-64 b \cosh (6 \text {arcsinh}(c x))-9 b \cosh (8 \text {arcsinh}(c x))-24 \text {arcsinh}(c x) (120 a+48 b \sinh (2 \text {arcsinh}(c x))-24 b \sinh (4 \text {arcsinh}(c x))-16 b \sinh (6 \text {arcsinh}(c x))-3 b \sinh (8 \text {arcsinh}(c x)))\right )}{73728 c^3} \] Input:

Integrate[x^2*(Pi + c^2*Pi*x^2)^(5/2)*(a + b*ArcSinh[c*x]),x]
 

Output:

(Pi^(5/2)*(2880*a*c*x*Sqrt[1 + c^2*x^2] + 22656*a*c^3*x^3*Sqrt[1 + c^2*x^2 
] + 26112*a*c^5*x^5*Sqrt[1 + c^2*x^2] + 9216*a*c^7*x^7*Sqrt[1 + c^2*x^2] - 
 1440*b*ArcSinh[c*x]^2 + 576*b*Cosh[2*ArcSinh[c*x]] - 144*b*Cosh[4*ArcSinh 
[c*x]] - 64*b*Cosh[6*ArcSinh[c*x]] - 9*b*Cosh[8*ArcSinh[c*x]] - 24*ArcSinh 
[c*x]*(120*a + 48*b*Sinh[2*ArcSinh[c*x]] - 24*b*Sinh[4*ArcSinh[c*x]] - 16* 
b*Sinh[6*ArcSinh[c*x]] - 3*b*Sinh[8*ArcSinh[c*x]])))/(73728*c^3)
 

Rubi [A] (verified)

Time = 1.97 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.19, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {6223, 243, 49, 2009, 6223, 244, 2009, 6221, 15, 6227, 15, 6198}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int x^2 \left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx\)

\(\Big \downarrow \) 6223

\(\displaystyle \frac {5}{8} \pi \int x^2 \left (c^2 \pi x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))dx-\frac {1}{8} \pi ^{5/2} b c \int x^3 \left (c^2 x^2+1\right )^2dx+\frac {1}{8} x^3 \left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x))\)

\(\Big \downarrow \) 243

\(\displaystyle \frac {5}{8} \pi \int x^2 \left (c^2 \pi x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))dx-\frac {1}{16} \pi ^{5/2} b c \int x^2 \left (c^2 x^2+1\right )^2dx^2+\frac {1}{8} x^3 \left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x))\)

\(\Big \downarrow \) 49

\(\displaystyle \frac {5}{8} \pi \int x^2 \left (c^2 \pi x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))dx-\frac {1}{16} \pi ^{5/2} b c \int \left (c^4 x^6+2 c^2 x^4+x^2\right )dx^2+\frac {1}{8} x^3 \left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x))\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {5}{8} \pi \int x^2 \left (c^2 \pi x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))dx+\frac {1}{8} x^3 \left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x))-\frac {1}{16} \pi ^{5/2} b c \left (\frac {c^4 x^8}{4}+\frac {2 c^2 x^6}{3}+\frac {x^4}{2}\right )\)

\(\Big \downarrow \) 6223

\(\displaystyle \frac {5}{8} \pi \left (\frac {1}{2} \pi \int x^2 \sqrt {c^2 \pi x^2+\pi } (a+b \text {arcsinh}(c x))dx-\frac {1}{6} \pi ^{3/2} b c \int x^3 \left (c^2 x^2+1\right )dx+\frac {1}{6} x^3 \left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))\right )+\frac {1}{8} x^3 \left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x))-\frac {1}{16} \pi ^{5/2} b c \left (\frac {c^4 x^8}{4}+\frac {2 c^2 x^6}{3}+\frac {x^4}{2}\right )\)

\(\Big \downarrow \) 244

\(\displaystyle \frac {5}{8} \pi \left (\frac {1}{2} \pi \int x^2 \sqrt {c^2 \pi x^2+\pi } (a+b \text {arcsinh}(c x))dx-\frac {1}{6} \pi ^{3/2} b c \int \left (c^2 x^5+x^3\right )dx+\frac {1}{6} x^3 \left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))\right )+\frac {1}{8} x^3 \left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x))-\frac {1}{16} \pi ^{5/2} b c \left (\frac {c^4 x^8}{4}+\frac {2 c^2 x^6}{3}+\frac {x^4}{2}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {5}{8} \pi \left (\frac {1}{2} \pi \int x^2 \sqrt {c^2 \pi x^2+\pi } (a+b \text {arcsinh}(c x))dx+\frac {1}{6} x^3 \left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {1}{6} \pi ^{3/2} b c \left (\frac {c^2 x^6}{6}+\frac {x^4}{4}\right )\right )+\frac {1}{8} x^3 \left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x))-\frac {1}{16} \pi ^{5/2} b c \left (\frac {c^4 x^8}{4}+\frac {2 c^2 x^6}{3}+\frac {x^4}{2}\right )\)

\(\Big \downarrow \) 6221

\(\displaystyle \frac {5}{8} \pi \left (\frac {1}{2} \pi \left (\frac {1}{4} \sqrt {\pi } \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx-\frac {1}{4} \sqrt {\pi } b c \int x^3dx+\frac {1}{4} x^3 \sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))\right )+\frac {1}{6} x^3 \left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {1}{6} \pi ^{3/2} b c \left (\frac {c^2 x^6}{6}+\frac {x^4}{4}\right )\right )+\frac {1}{8} x^3 \left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x))-\frac {1}{16} \pi ^{5/2} b c \left (\frac {c^4 x^8}{4}+\frac {2 c^2 x^6}{3}+\frac {x^4}{2}\right )\)

\(\Big \downarrow \) 15

\(\displaystyle \frac {5}{8} \pi \left (\frac {1}{2} \pi \left (\frac {1}{4} \sqrt {\pi } \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx+\frac {1}{4} x^3 \sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))-\frac {1}{16} \sqrt {\pi } b c x^4\right )+\frac {1}{6} x^3 \left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {1}{6} \pi ^{3/2} b c \left (\frac {c^2 x^6}{6}+\frac {x^4}{4}\right )\right )+\frac {1}{8} x^3 \left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x))-\frac {1}{16} \pi ^{5/2} b c \left (\frac {c^4 x^8}{4}+\frac {2 c^2 x^6}{3}+\frac {x^4}{2}\right )\)

\(\Big \downarrow \) 6227

\(\displaystyle \frac {5}{8} \pi \left (\frac {1}{2} \pi \left (\frac {1}{4} \sqrt {\pi } \left (-\frac {\int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}dx}{2 c^2}-\frac {b \int xdx}{2 c}+\frac {x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{2 c^2}\right )+\frac {1}{4} x^3 \sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))-\frac {1}{16} \sqrt {\pi } b c x^4\right )+\frac {1}{6} x^3 \left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {1}{6} \pi ^{3/2} b c \left (\frac {c^2 x^6}{6}+\frac {x^4}{4}\right )\right )+\frac {1}{8} x^3 \left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x))-\frac {1}{16} \pi ^{5/2} b c \left (\frac {c^4 x^8}{4}+\frac {2 c^2 x^6}{3}+\frac {x^4}{2}\right )\)

\(\Big \downarrow \) 15

\(\displaystyle \frac {5}{8} \pi \left (\frac {1}{2} \pi \left (\frac {1}{4} \sqrt {\pi } \left (-\frac {\int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}dx}{2 c^2}+\frac {x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{2 c^2}-\frac {b x^2}{4 c}\right )+\frac {1}{4} x^3 \sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))-\frac {1}{16} \sqrt {\pi } b c x^4\right )+\frac {1}{6} x^3 \left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {1}{6} \pi ^{3/2} b c \left (\frac {c^2 x^6}{6}+\frac {x^4}{4}\right )\right )+\frac {1}{8} x^3 \left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x))-\frac {1}{16} \pi ^{5/2} b c \left (\frac {c^4 x^8}{4}+\frac {2 c^2 x^6}{3}+\frac {x^4}{2}\right )\)

\(\Big \downarrow \) 6198

\(\displaystyle \frac {1}{8} x^3 \left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x))+\frac {5}{8} \pi \left (\frac {1}{6} x^3 \left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {1}{2} \pi \left (\frac {1}{4} x^3 \sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))+\frac {1}{4} \sqrt {\pi } \left (-\frac {(a+b \text {arcsinh}(c x))^2}{4 b c^3}+\frac {x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{2 c^2}-\frac {b x^2}{4 c}\right )-\frac {1}{16} \sqrt {\pi } b c x^4\right )-\frac {1}{6} \pi ^{3/2} b c \left (\frac {c^2 x^6}{6}+\frac {x^4}{4}\right )\right )-\frac {1}{16} \pi ^{5/2} b c \left (\frac {c^4 x^8}{4}+\frac {2 c^2 x^6}{3}+\frac {x^4}{2}\right )\)

Input:

Int[x^2*(Pi + c^2*Pi*x^2)^(5/2)*(a + b*ArcSinh[c*x]),x]
 

Output:

-1/16*(b*c*Pi^(5/2)*(x^4/2 + (2*c^2*x^6)/3 + (c^4*x^8)/4)) + (x^3*(Pi + c^ 
2*Pi*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/8 + (5*Pi*(-1/6*(b*c*Pi^(3/2)*(x^4/4 
 + (c^2*x^6)/6)) + (x^3*(Pi + c^2*Pi*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/6 + 
(Pi*(-1/16*(b*c*Sqrt[Pi]*x^4) + (x^3*Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[ 
c*x]))/4 + (Sqrt[Pi]*(-1/4*(b*x^2)/c + (x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh 
[c*x]))/(2*c^2) - (a + b*ArcSinh[c*x])^2/(4*b*c^3)))/4))/2))/8
 

Defintions of rubi rules used

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -1]
 

rule 49

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int 
[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] 
&& IGtQ[m, 0] && IGtQ[m + n + 2, 0]
 

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

rule 244

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand 
Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p 
, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 6198

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_ 
Symbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]]*( 
a + b*ArcSinh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c 
^2*d] && NeQ[n, -1]
 

rule 6221

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*Sqrt[(d_) + 
 (e_.)*(x_)^2], x_Symbol] :> Simp[(f*x)^(m + 1)*Sqrt[d + e*x^2]*((a + b*Arc 
Sinh[c*x])^n/(f*(m + 2))), x] + (Simp[(1/(m + 2))*Simp[Sqrt[d + e*x^2]/Sqrt 
[1 + c^2*x^2]]   Int[(f*x)^m*((a + b*ArcSinh[c*x])^n/Sqrt[1 + c^2*x^2]), x] 
, x] - Simp[b*c*(n/(f*(m + 2)))*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]]   I 
nt[(f*x)^(m + 1)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d 
, e, f, m}, x] && EqQ[e, c^2*d] && IGtQ[n, 0] && (IGtQ[m, -2] || EqQ[n, 1])
 

rule 6223

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ 
.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^p*((a + b*Arc 
Sinh[c*x])^n/(f*(m + 2*p + 1))), x] + (Simp[2*d*(p/(m + 2*p + 1))   Int[(f* 
x)^m*(d + e*x^2)^(p - 1)*(a + b*ArcSinh[c*x])^n, x], x] - Simp[b*c*(n/(f*(m 
 + 2*p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p]   Int[(f*x)^(m + 1)*(1 + 
c^2*x^2)^(p - 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, 
 d, e, f, m}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && GtQ[p, 0] &&  !LtQ[m, -1]
 

rule 6227

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ 
.)*(x_)^2)^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a 
+ b*ArcSinh[c*x])^n/(e*(m + 2*p + 1))), x] + (-Simp[f^2*((m - 1)/(c^2*(m + 
2*p + 1)))   Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, x], x] 
 - Simp[b*f*(n/(c*(m + 2*p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p]   Int 
[(f*x)^(m - 1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x] 
) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && IGtQ[ 
m, 1] && NeQ[m + 2*p + 1, 0]
 

Maple [A] (verified)

Time = 1.06 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.26

method	result	size
default	\(\frac {a x \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {7}{2}}}{8 \pi \,c^{2}}-\frac {a x \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {5}{2}}}{48 c^{2}}-\frac {5 a \pi x \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}{192 c^{2}}-\frac {5 a \,\pi ^{2} x \sqrt {\pi \,c^{2} x^{2}+\pi }}{128 c^{2}}-\frac {5 a \,\pi ^{3} \ln \left (\frac {\pi \,c^{2} x}{\sqrt {\pi \,c^{2}}}+\sqrt {\pi \,c^{2} x^{2}+\pi }\right )}{128 c^{2} \sqrt {\pi \,c^{2}}}-\frac {b \,\pi ^{\frac {5}{2}} \left (-288 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x^{7} c^{7}+36 c^{8} x^{8}-816 \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right ) x^{5} c^{5}+136 c^{6} x^{6}-708 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}+177 c^{4} x^{4}-90 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x c +45 c^{2} x^{2}+45 \operatorname {arcsinh}\left (x c \right )^{2}-32\right )}{2304 c^{3}}\)	\(267\)
parts	\(\frac {a x \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {7}{2}}}{8 \pi \,c^{2}}-\frac {a x \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {5}{2}}}{48 c^{2}}-\frac {5 a \pi x \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}{192 c^{2}}-\frac {5 a \,\pi ^{2} x \sqrt {\pi \,c^{2} x^{2}+\pi }}{128 c^{2}}-\frac {5 a \,\pi ^{3} \ln \left (\frac {\pi \,c^{2} x}{\sqrt {\pi \,c^{2}}}+\sqrt {\pi \,c^{2} x^{2}+\pi }\right )}{128 c^{2} \sqrt {\pi \,c^{2}}}-\frac {b \,\pi ^{\frac {5}{2}} \left (-288 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x^{7} c^{7}+36 c^{8} x^{8}-816 \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right ) x^{5} c^{5}+136 c^{6} x^{6}-708 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}+177 c^{4} x^{4}-90 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x c +45 c^{2} x^{2}+45 \operatorname {arcsinh}\left (x c \right )^{2}-32\right )}{2304 c^{3}}\)	\(267\)

Input:

int(x^2*(Pi*c^2*x^2+Pi)^(5/2)*(a+b*arcsinh(x*c)),x,method=_RETURNVERBOSE)
 

Output:

1/8*a*x*(Pi*c^2*x^2+Pi)^(7/2)/Pi/c^2-1/48*a/c^2*x*(Pi*c^2*x^2+Pi)^(5/2)-5/ 
192*a/c^2*Pi*x*(Pi*c^2*x^2+Pi)^(3/2)-5/128*a/c^2*Pi^2*x*(Pi*c^2*x^2+Pi)^(1 
/2)-5/128*a/c^2*Pi^3*ln(Pi*c^2*x/(Pi*c^2)^(1/2)+(Pi*c^2*x^2+Pi)^(1/2))/(Pi 
*c^2)^(1/2)-1/2304*b*Pi^(5/2)*(-288*arcsinh(x*c)*(c^2*x^2+1)^(1/2)*x^7*c^7 
+36*c^8*x^8-816*(c^2*x^2+1)^(1/2)*arcsinh(x*c)*x^5*c^5+136*c^6*x^6-708*arc 
sinh(x*c)*(c^2*x^2+1)^(1/2)*x^3*c^3+177*c^4*x^4-90*arcsinh(x*c)*(c^2*x^2+1 
)^(1/2)*x*c+45*c^2*x^2+45*arcsinh(x*c)^2-32)/c^3
 

Fricas [F]

\[ \int x^2 \left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\int { {\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {5}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{2} \,d x } \] Input:

integrate(x^2*(pi*c^2*x^2+pi)^(5/2)*(a+b*arcsinh(c*x)),x, algorithm="frica 
s")
 

Output:

integral(sqrt(pi + pi*c^2*x^2)*(pi^2*a*c^4*x^6 + 2*pi^2*a*c^2*x^4 + pi^2*a 
*x^2 + (pi^2*b*c^4*x^6 + 2*pi^2*b*c^2*x^4 + pi^2*b*x^2)*arcsinh(c*x)), x)
 

Sympy [A] (verification not implemented)

Time = 48.10 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.65 \[ \int x^2 \left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\begin {cases} \frac {\pi ^{\frac {5}{2}} a c^{4} x^{7} \sqrt {c^{2} x^{2} + 1}}{8} + \frac {17 \pi ^{\frac {5}{2}} a c^{2} x^{5} \sqrt {c^{2} x^{2} + 1}}{48} + \frac {59 \pi ^{\frac {5}{2}} a x^{3} \sqrt {c^{2} x^{2} + 1}}{192} + \frac {5 \pi ^{\frac {5}{2}} a x \sqrt {c^{2} x^{2} + 1}}{128 c^{2}} - \frac {5 \pi ^{\frac {5}{2}} a \operatorname {asinh}{\left (c x \right )}}{128 c^{3}} - \frac {\pi ^{\frac {5}{2}} b c^{5} x^{8}}{64} + \frac {\pi ^{\frac {5}{2}} b c^{4} x^{7} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{8} - \frac {17 \pi ^{\frac {5}{2}} b c^{3} x^{6}}{288} + \frac {17 \pi ^{\frac {5}{2}} b c^{2} x^{5} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{48} - \frac {59 \pi ^{\frac {5}{2}} b c x^{4}}{768} + \frac {59 \pi ^{\frac {5}{2}} b x^{3} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{192} - \frac {5 \pi ^{\frac {5}{2}} b x^{2}}{256 c} + \frac {5 \pi ^{\frac {5}{2}} b x \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{128 c^{2}} - \frac {5 \pi ^{\frac {5}{2}} b \operatorname {asinh}^{2}{\left (c x \right )}}{256 c^{3}} & \text {for}\: c \neq 0 \\\frac {\pi ^{\frac {5}{2}} a x^{3}}{3} & \text {otherwise} \end {cases} \] Input:

integrate(x**2*(pi*c**2*x**2+pi)**(5/2)*(a+b*asinh(c*x)),x)
 

Output:

Piecewise((pi**(5/2)*a*c**4*x**7*sqrt(c**2*x**2 + 1)/8 + 17*pi**(5/2)*a*c* 
*2*x**5*sqrt(c**2*x**2 + 1)/48 + 59*pi**(5/2)*a*x**3*sqrt(c**2*x**2 + 1)/1 
92 + 5*pi**(5/2)*a*x*sqrt(c**2*x**2 + 1)/(128*c**2) - 5*pi**(5/2)*a*asinh( 
c*x)/(128*c**3) - pi**(5/2)*b*c**5*x**8/64 + pi**(5/2)*b*c**4*x**7*sqrt(c* 
*2*x**2 + 1)*asinh(c*x)/8 - 17*pi**(5/2)*b*c**3*x**6/288 + 17*pi**(5/2)*b* 
c**2*x**5*sqrt(c**2*x**2 + 1)*asinh(c*x)/48 - 59*pi**(5/2)*b*c*x**4/768 + 
59*pi**(5/2)*b*x**3*sqrt(c**2*x**2 + 1)*asinh(c*x)/192 - 5*pi**(5/2)*b*x** 
2/(256*c) + 5*pi**(5/2)*b*x*sqrt(c**2*x**2 + 1)*asinh(c*x)/(128*c**2) - 5* 
pi**(5/2)*b*asinh(c*x)**2/(256*c**3), Ne(c, 0)), (pi**(5/2)*a*x**3/3, True 
))
 

Maxima [F(-2)]

Exception generated. \[ \int x^2 \left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate(x^2*(pi*c^2*x^2+pi)^(5/2)*(a+b*arcsinh(c*x)),x, algorithm="maxim 
a")
 

Output:

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati 
ve exponent.
 

Giac [F]

\[ \int x^2 \left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\int { {\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {5}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{2} \,d x } \] Input:

integrate(x^2*(pi*c^2*x^2+pi)^(5/2)*(a+b*arcsinh(c*x)),x, algorithm="giac" 
)
 

Output:

integrate((pi + pi*c^2*x^2)^(5/2)*(b*arcsinh(c*x) + a)*x^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int x^2 \left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\int x^2\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (\Pi \,c^2\,x^2+\Pi \right )}^{5/2} \,d x \] Input:

int(x^2*(a + b*asinh(c*x))*(Pi + Pi*c^2*x^2)^(5/2),x)
 

Output:

int(x^2*(a + b*asinh(c*x))*(Pi + Pi*c^2*x^2)^(5/2), x)
 

Reduce [F]

\[ \int x^2 \left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {\sqrt {\pi }\, \pi ^{2} \left (48 \sqrt {c^{2} x^{2}+1}\, a \,c^{7} x^{7}+136 \sqrt {c^{2} x^{2}+1}\, a \,c^{5} x^{5}+118 \sqrt {c^{2} x^{2}+1}\, a \,c^{3} x^{3}+15 \sqrt {c^{2} x^{2}+1}\, a c x +384 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x^{6}d x \right ) b \,c^{7}+768 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x^{4}d x \right ) b \,c^{5}+384 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x^{2}d x \right ) b \,c^{3}-15 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x \right ) a \right )}{384 c^{3}} \] Input:

int(x^2*(Pi*c^2*x^2+Pi)^(5/2)*(a+b*asinh(c*x)),x)
 

Output:

(sqrt(pi)*pi**2*(48*sqrt(c**2*x**2 + 1)*a*c**7*x**7 + 136*sqrt(c**2*x**2 + 
 1)*a*c**5*x**5 + 118*sqrt(c**2*x**2 + 1)*a*c**3*x**3 + 15*sqrt(c**2*x**2 
+ 1)*a*c*x + 384*int(sqrt(c**2*x**2 + 1)*asinh(c*x)*x**6,x)*b*c**7 + 768*i 
nt(sqrt(c**2*x**2 + 1)*asinh(c*x)*x**4,x)*b*c**5 + 384*int(sqrt(c**2*x**2 
+ 1)*asinh(c*x)*x**2,x)*b*c**3 - 15*log(sqrt(c**2*x**2 + 1) + c*x)*a))/(38 
4*c**3)